ScotCats 9: Fibrations in Computation Workshop

The idea of the workshop is to get people using fibrational methods
in computer science together to catch up with what each of us is
doing, to learn from each others techniques and to possibly chart
ideas for future development and interaction.

Abstracts

Bisimulation up-to enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain up-to techniques not only for bisimilarity but for a large class of coinductive predicates modelled as coalgebras. By tuning the parameters of our framework, we obtain novel techniques for unary predicates and nominal automata, a variant of the GSOS rule format for similarity, and a new categorical treatment of weak bisimilarity.

Let CC be a category with finite limits. Moens' theorem
asserts an equivalence between the 2-category
ExFib(CC) of extensive fibrations on CC, and
the pseudo-co-slice category CC/Lex of finite-limit categories
under CC. Under this equivalence, the fibration correspdonding to
a functor F: CC → DD is the pullback of the codomain fibration
of DD along FF, called gluing fibration of F.

In my talk, I want to explain how sheaf constructions and
Grothendieck toposes relative to a base topos SS can be
understood in terms of gluing fibrations and Moens' theorem,
and how to generalize this framework to include realizability
toposes.

Then I will describe how this point of view on realizability
toposes allows to view them as cocompletions of generalized
preorders (analogous to presheaves on meet-semilattices)
and state an intrinsic characterization of realizability
toposes over partial combinatory algebras, which arises
quite naturally in this framework.

Motivated by game semantics, we construct a fibration from a `double factorisation system' (Tholen and Pultr) satisfying a few additional hypotheses. We construct a (pseudo) double category, using cospans whose legs satisfy certain factorisation properties. We show that the `vertical codomain' functor of this double category is then a fibration.
Directed graphs will be used throughout as a stripped down example of a game.

In this talk I will give categorical semantics for
Andrew Kennedy’s Units of Measure (1998). I'll start by following the
standard approach in categorical logical and
type theory by using a fibrational structure to separate
index-information (the units) from indexed-information (the types which
may contain units). This will lead us to the notion of a UoM-fibration.
I'll give examples of UoM fibrations, including a model based upon
G-sets, and theorems allowing the construction of further
UoM-fibrations. Finally, I'll touch on parametricity in this setting.

In this talk we will discuss a new application of Grothendieck bifibrations in algebra. It relies on considering the bifibration of substructures of familiar group-like structures and reformulating standard results such as isomorphism theorems and diagram lemmas in terms of this bifibration, after which a completely self-dual approach to these results becomes possible using the intrinsic duality of the bifibrational language. This easily extends to categorical axiomatic contexts (where, as expected, substructures are traded for subobjects) studied in categorical and universal algebra, such as semi-abelian categories. Moreover, it becomes possible to characterise these contexts by dual axioms on the bifibration of subobjects [1]. We will then explain that one of the main ingredients of the theory is an additional 2-dimensional simplicial structure hidden behind the bifibration of subobjects in these categories. This work answers the proposal of Mac Lane in his paper Duality for Groups [4], namely that isomorphism theorems and similar results for groups be established in a self-dual categorical context. Pre-prints of two papers on this work are available [3,4].

Accommodation

From Glasgow Queen Street Station or Glasgow Central Station

Livingstone Tower is a few minutes walk from Queen Street Station in
the centre of Glasgow, and about 10 minutes walk from Central
Station. The map above shows the general area, as well as
recommended walking routes.

From Edinburgh Airport

For people arriving
via Edinburgh
Airport, the best route is to take
the airport bus to Buchanan Street Bus station in Glasgow, and then to walk or take a taxi
from there. There are buses every 30 minutes during the day,
and every hour after 19.30. The bus costs £16 for an open return.
Here are general instructions on travel to and from Edinburgh airport.